Successful implementation of Bayesian methods in wider circumstances than presently available promises to substantially enhance analyses of data from clinical trials and other longitudinal studies. The work proposed here concerns implementation and extension of Bayesian techniques as outlined in a recent discussion by Greenhouse (1990) of the NCCTG Clinical Trial of Advanced Colorectal Carcinoma. Although in this proposal we confine ourselves mainly to methodological issues, our basic objective is to provide appropriate methods for advancing relevant areas of applications in the health and biomedical sciences. To this end, we attempt to motivate the methodological problems that we wish to solve with examples from specific clinical trials in cancer, in lung disease, and in mental health. Our research agenda can be divided into three main areas: model specification, assessing sensitivity of inferences to prior distributions, and computing. In each area, hierarchical statistical models will be given special attention. Our research interests in model specification are threefold and include: (1) Bayesian hypothesis testing and model selection, (2) the problem of reparameterizing a model, and (3) hierarchical models with nonparametric Dirichlet process priors. Our second main area of interest is the assessment of the sensitivity of the posterior distribution to the choice of prior. We will investigate the use of computational tools to make robust Bayesian inference practical, both in the parametric and nonparametric setting, and we will apply these techniques to the problem of assessing sensitivity in hierarchical models. Finally, since the implementation of Bayesian methods requires extensive computing, we will continue to study the use of Laplace's method for approximating posterior distributions in the case of specific, complex models, such as mixtures of survival distributions. We will also investigate the use and improvement of Gibbs sampling in applications to inferential problems in the health and biomedical sciences.